We present a conceptual design of a conduction-cooled 15 T/220 mm warm-bore superconducting magnet for use in a 350 GHz, 1 MW gyrotron. Gyrotrons can generate MW class of continuous wave power per device in the millimeter wave band and thus make them suitable for heating plasmas and driving current in magnetically confined fusion devices. The 12 T class tokamaks require a heating frequency of ∼350 GHz and a power level of 1-2 MW per gyrotron, as part of a 20-40 MW electron cyclotron resonance heating system. Gyrotrons at 350 GHz and 1 MW require superconducting magnets with ∼15 T field with a specific field profile and a warm bore diameter of 220 mm. In this paper, we propose to use high-temperature superconducting REBCO magnet technology in the magnet design, with a focus on managing stress, and implementing highly reliable winding and quench protection methods. We also propose to use an actively shielded low-temperature superconducting NbTi magnet in the lower field region to cost-effectively contribute a ∼5 T field and reduce the fringe field.